A Radical Approach to Lebesgue's Theory of Integration

by David M. Bressoud

This lively introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems, highlighting the difficulties mathematicians encountered as these ideas were refined. The story begins with Riemann’s definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemann’s definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis.

Genres: Mathematics

344 pages, Paperback

First published January 1, 2008

Reviews

[Kevin Doran · Rating 1 out of 5]

Didn't finish.

Did not like it. Nice idea but terrible execution.

Go read Stephen Abbott's "Understanding Analysis" for a book that does a good job of showing the reader why the subject has evolved in the way it has.

The most serious flaw of "A radical approach...." is that there isn't a coherent progression. Often when starting a new section it feels like "oh, I guess we are talking about this now". It feels like a whole chapter is just a bullet-point list. I feel myself flicking to the end of a chapter to figure out what concept is being approached.

The minor flaws add up too. When reading, I feel like I need to get over so many unnecessary obstacles to read the book. Some examples:
- The notation used in the book is described after the introduction, which makes it hard to refer back to.
- Inconsistent headings. For example, section 2.3 introduces relevant work of 3 mathematicians; the first two are described in detail under the "Section 2.3" heading, while the 3rd mathematician is given his own sub-heading.
- The headings use the 1.2.3.4 style; however, this style is only used for the first 2 levels, and the third section has no numbering. This is quite a minor point, but it just adds to the ease at which one can get lost in a chapter.
- Definitions of quantities appear in separate floating boxes on a page. The other text continues around the box and you only realize that you need to read the box once you stumble across some number or function that hasn't yet been defined. This process is really jarring as you need to bring all the required variables back to into working memory in order to resume reading.
- The mini personal histories of the mathematicians often break the flow of the mathematical discussion. Instead of the definitions, maybe the personal histories would be better in the boxes that are separated from the normal text flow.
- The proofs are hard to read. It's as if the author decided to avoid using any supporting conclusions in the form of lemmas, and instead he just pushes these extra argument steps into the beginning of a proof. In my opinion, if you are proofing an if-X-then-Y statement, you should start with an "assume X", not a separate proof of something else that is used later. The author also makes statements like "it then follows that...", without specifying what enables this implication.

[Adam · Rating 5 out of 5]

After reading this book, I've become convinced that every book on a technical subject should be written this way -- with motivation and historical context which keeps the reader thinking about why and just what this is all about.

[Nick Black · Rating 3 out of 5]

Amazon 2009-04-12. Ahhh, the Lebesgue integral, fearsome opponent of advanced undergraduates everywhere and the first great real departure (if you'll forgive the pun, haha) from high-school mathematics in the standard university curriculum (usually the focus of a second or third semester in real analysis, and almost a necessary prerequisite for functional analysis and study of topological vector spaces (especially the Banach and Hilbert spaces so central to quantum mechanics)). The modern function maps a set (domain) to a set (range); the Lebesgue measure defines "length" (with its natural additivity and translation invariance properties) for sets more complex than those addressed by Riemann -- the domains of the countably additive functions necessitated by modern probability theory, the unbounded sets giving rise to "improper" Riemann integrals, the limits of sequences of functions that underpin nowhere-continuity. Using a Lebesgue measure space (a set, the σ-algebra of Lebesgue-measurable subsets thereof, and the measure itself). The integral is now built up, rather than the Riemannian sum of products of values and infinitesimal subsections (f(x)*(dx)), as the more general sum of products of values, measure and infinitesimal subsets (f(x)*(dμ) == f(x)*μ(dx)).

Whew.

Unfortunately, I don't really understand any of this at a deep, daily-workable level (then again, how often do continuous probabilities really come up in computing? unfortunately, sequences of functions spring up like digitaria sanguinalis among every field into which I dip my wick (can we say Fast Fourier Transform, boys and girls (mainly boys, sigh)))? But...that's why we read the book. I wish mighty Bartle's The Elements of Real Analysis covered advanced integration, but it pretty much throws in the towel following rigorous coverage of Riemann-Stieltjes, zorn.